In deep learning, the aleatoric uncertainty expressed through \(\sigma _y\)  is often given a greater importance and is regarded as measuring inherent, irreducible label ambiguity [1]}, [2]}, [3]}, [4]}.
However, such a point of view does no longer apply in those regression problems where the goal is the prediction of the regression function \(f_\theta (x^*)\)  at some \(x^*\) . In this case only the epistemic uncertainty about the network's parameter \(\theta \)  remains which, in principle, could become arbitrarily small as the training set grows.
Furthermore, typically \(x^*=\zeta ^*+\varepsilon _x\)  contains some noise \(\varepsilon _x\)  as well, and one is actually interested in the value of \(f_\theta (\zeta ^\ast )\) .
The uncertainty caused by the fact that \(x^*\) , and not \(\zeta ^*\) , is observed constitutes an aleatoric part of the uncertainty for the prediction of \(f_\theta (\zeta ^*)\)  which is neither covered by \(\sigma _y\)  nor by the uncertainty of \(\theta \) .
